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Journal of Financial Economics 81 (2006) 143–173
On the role of arbitrageurs in rational markets$
Suleyman Basaka, Benjamin Croitorub,Ã
aLondon Business School and CEPR, Regent’s Park, London NW1 4SA, UK bMcGill University, 1001 Sherbrooke Street West, Montreal, Quebec, H3A 1G5 Canada
Received 30 September 2003; received in revised form 29 April 2004; accepted 11 November 2004
Available online 10 January 2006
Abstract
We investigate the role of ‘‘arbitrageurs,’’ who exploit price discrepancies between redundant
securities. Arbitrage opportunities arise endogenously in an economy populated by rational,
heterogeneous investors facing investment restrictions. We show that an arbitrageur alleviates these
restrictions and improves the transfer of risk amongst investors. When the arbitrageur behaves
noncompetitively, taking into account the price impact of his trades, he optimally limits the size of his
positions due to his decreasing marginal profits. When the arbitrageur is subject to marginrequirements and is endowed with capital from outside investors, the size of his trades and capital are
endogenously determined in equilibrium.
r 2005 Published by Elsevier B.V.
JEL classification: C60; D50; D90; G11; G12
Keywords: Arbitrage; Asset pricing; Margin requirements; Noncompetitive markets; Risk-sharing
ARTICLE IN PRESS
www.elsevier.com/locate/jfec
0304-405X/$ - see front matter r 2005 Published by Elsevier B.V.
doi:10.1016/j.jfineco.2004.11.004
$We are grateful to an anonymous referee for valuable suggestions and to Viral Acharya, Domenico Cuoco,
Michael Gallmeyer, Wei Xiong, seminar participants at the London School of Economics, McGill University, the
University of Vienna, the University of Wisconsin-Madison, the 2003 GERAD Optimization Days, the Blaise
Pascal International Conference on Financial Modeling, the 2004 American Finance Association Meetings, the
2004 European Finance Association Meetings and the 2005 Isaac Newton Institute Workshop on Mathematical
Finance for their helpful comments. Financial support from the Economic and Social Research Council for the
first author and from the Fonds Que ´ becois de Recherche sur la Socie ´ te ´ et la Culture, the Institut de Finance
Mathe ´ matique de Montre ´ al and the Social Sciences and Humanities Research Council of Canada for the second
author is gratefully acknowledged. All errors are solely our responsibility.Ã
Corresponding author. Fax: +1 514 398 3876.E-mail address: [email protected] (B. Croitoru).
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1. Introduction
The presence of apparent inconsistencies in asset prices has long been well documented.
Over the years, various types of securities such as primes and scores, stock index futures,
closed-end funds have been found to consistently deviate from their no-arbitrage values.Recent examples include the deviations from put-call parity in options markets (Ofek
et al., 2004) and the paradoxical behavior of prices in some equity carve-outs ( Lamont and
Thaler, 2003). For example, in the case of 3Com and Palm in March of 2000, the market
value of 3Com, net of its holdings of Palm, became negative on Palm’s first day of trading.
Such mispricings clearly lead to opportunities for arbitrage profits, and not surprisingly,
there is ample evidence of market participants engaging in trades designed to reap these
profits. For example, one type of arbitrage transaction, stock index arbitrage, is estimated
by the NYSE (www.nyse.com) to typically make up 7% to 9% of the program trading
volume, which itself represents about one-third of total NYSE trading, and many other
instruments, such as bonds and derivatives, are also frequently involved in arbitrages.
Moreover, there is also considerable anecdotal evidence of some investors specializing in
such trades. As Shleifer (2000, Chapter 4) puts it, ‘‘commonly, arbitrage is conducted by
relatively few, highly specialized investors.’’ A key example of such specialization is that of
hedge funds, many of which focus on arbitrage strategies (Lhabitant, 2002). Their
economic role, however, is highly controversial (especially after the infamous collapse of
the Long Term Capital Management hedge fund in 1998), underscoring the importance of
a rigorous study of the impact of arbitrage.
A growing thread of the academic literature studies these phenomena. Much of this
literature (see Shleifer, 2000, for a survey) has postulated the presence of irrational noisetraders whose erratic behavior generates mispricings. In such work, the term arbitrageur
typically refers to a rational trader, as opposed to irrational noise traders.
In this paper, we employ a different approach that is little explored in the literature so
far. We present an equilibrium in which all market participants are rational. Instead of
resorting to irrational noise traders to generate arbitrage opportunities, in our model these
arise endogenously in equilibrium, following from the presence of heterogeneous rational
investors subject to investment restrictions such as leverage constraints or restrictions on
short sales.1 Irrespective of whether or not noise traders are present in actual financial
markets, our view is that there may exist different categories of rational market
participants (e.g., long-term pension funds and short-term hedge funds) whose interactionsare sufficiently rich in implications. Our arbitrageurs are specialized traders who only take
on riskless, costless arbitrage positions. Our goal is to study the equilibrium impact of the
presence of these arbitrageurs. More specifically, we consider an economy with two
heterogeneous, rational, risk-taking investors and we examine how the presence of an
arbitrageur affects the way in which risk is traded between the investors, as well as the
ensuing equilibrium prices and consumption allocations. The investors could represent
real-life retail investors or mutual funds, while the arbitrageur in our model could be an
ARTICLE IN PRESS
1Recent empirical and theoretical work (e.g., Duffie et al., 2002; Ofek et al., 2004) points to impediments to
short-selling comparable to our investment restrictions as a possible cause for the presence of arbitrageopportunities. The restrictions on short-selling in Duffie et al. (2002) can be thought of as stochastic constraints
that could be incorporated into our work without considerably affecting our main message. Lamont and Thaler
(2003) emphasize the role of short-sale constraints, a part of the restrictions we assume, in making mispricings
possible.
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arbitrage desk at an investment bank or a hedge fund that specializes in arbitrage
strategies. Our setup is comparable to the main workhorse models in asset pricing ( Lucas,
1978; Cox, Ingersoll and Ross, 1985), which provides the added benefit of having well-
understood benchmarks for our results.
Since our main message is not dependent on the particular modeling setup employed, wefirst convey our insights in the simplest possible framework, namely, a single-period
economy with a finite set of states of nature and elementary securities (paying off in a
single state) available for trading, and redundant securities available in at least one state.
In this setting, ‘‘mispricing’’ means that securities with identical payoffs (here, elementary
securities paying off in the same state) trade at different prices, thereby generating a
riskless arbitrage opportunity. Without making any particular assumptions on the
investors, apart from the fact that their holdings of risky securities are constrained, we
show that the presence of an arbitrageur makes it possible for the investors to trade with
each other more and thus partially circumvent the investment restrictions.
The intuition is as follows. Suppose that one investor desires a much higher payoff than
the other. The first investor will then purchase securities from the latter investor. If the
discrepancy across investors is sufficiently large, the investment restrictions will kick in and
prevent investors from attaining their desired payoffs. At this point, a role for the
arbitrageur comes into play: The arbitrageur purchases securities from the investor who
desires the lower payoff, and converts the securities into a position in the redundant
security, which is then resold at a higher price to the other investor. The larger the
arbitrageur’s trades, the more he relieves the effects of the constraints. Thus, even though
he does not take on any risk, the arbitrageur facilitates risk-sharing amongst investors. The
difference between the arbitrageur’s ask and bid prices is the amount of the mispricing andthe source of his arbitrage profit.
To further explore the implications of the arbitrageur’s presence, we consider a richer
continuous-time setup in which investment opportunities include a risky stock, a riskless
bond, and a risky derivative financial security that is perfectly correlated with the stock.
Real-life examples that are particularly close to the derivative in our model include a stock
index futures contract, or a total return swap on a stock index. To make our model more
concrete, the latter example is developed in some length; however, our model could
accommodate any redundant, zero-net-supply risky security, and the actual nature of the
derivative does not significantly affect our results. For simplicity, the derivative is assumed
to have the same exposure to risk (as measured by volatility) as the stock. ‘‘Mispricing’’then refers to any discrepancy in the mean returns offered by the risky investments (stock
and derivative). Because these risky investments are redundant, such a pricing
inconsistency generates a riskless arbitrage opportunity that can be exploited by short-
selling the risky security with the lower mean return and investing the proceeds in the
other, generating a profit proportional to the difference in mean returns. For tractability,
our investors have logarithmic utility preferences. Moreover, the investors face two
investment restrictions, a leverage constraint that prevents them from taking very large
positions in the stock, and a restriction that limits short positions in the derivative. These
restrictions may represent margin regulations imposed by an exchange or an intermediary,
as well as collateral requirements imposed by a counterparty. To generate heterogeneityand trading activity, we assume that the investors disagree on the mean stock return; the
more optimistic investor desires a higher exposure to risk and accordingly buys risky assets
from the more pessimistic investor.
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Mispricing is shown to occur when the disagreement between investors is sufficiently
pronounced. Absent the mispricing, the more optimistic agent, who is prevented from
investing more in the stock by his leverage constraint, would demand more derivative than
what the pessimist can sell (without violating his short-sale limit on the derivative). The
mispricing, however, makes the derivative less attractively priced and thus reduces theoptimist’s demand to a level compatible with the pessimist’s constraints, i.e., allows
markets to clear.
Our intuition on the arbitrageur’s role from the single-period example readily extends to
the continuous-time case. When the disagreement is large enough for mispricing to occur,
the binding investment restrictions prevent the investors from trading as much as they
would like. To take advantage of the arbitrage opportunity that is available under
mispricing, the arbitrageur buys stock (from the pessimist) and sells derivatives (to the
optimist). This allows the pessimist to decrease his exposure to risk and the optimist to
increase his exposure to risk, which helps both investors achieve risk exposures closer to
those that they would optimally demand in the absence of investment restrictions. Thus, by
taking advantage of the arbitrage opportunity, the arbitrageur allows the investors to trade
more risk, effectively relieving the investors of their investment restrictions. The greater the
size of the arbitrageur’s trades, the greater this effect: As a result, the size of the mispricing
is decreasing in the arbitrageur’s position. We further demonstrate that the arbitrageur
plays a specific role that is different from that of an additional investor: While the
arbitrageur always improves risk-sharing among investors, a third investor could either
improve or impair it, depending on his beliefs.
For the greater part of the analysis, we assume that the arbitrageur behaves
competitively and faces a pre-specified position limit, which may be linked to a bank’scapital requirements or may simply represent an internal position limit. Then, under
mispricing, the arbitrageur always takes on the largest arbitrage position allowed by his
position limit. Hence, the size of his position is exogenous. To be able to endogenize the
arbitrageur’s position size, we provide some extensions of our basic model. First, we
consider the case in which the arbitrageur is noncompetitive, in that he takes into account
the impact of his trades on equilibrium prices so as to maximize his profit. In addition to
generating much richer arbitrageur behavior, the noncompetitive assumption may be more
realistic in the case of arbitrageurs that in real life are typically large, few in number,
specialized, and sophisticated. Due to his decreasing marginal profit, the noncompetitive
arbitrageur finds it optimal to limit the size of his trades. This makes the occurrence of mispricing more likely, because the noncompetitive arbitrageur will never trade enough
to fully ‘‘arbitrage away’’ the mispricing and make it disappear (driving his profits to zero).
While our main intuition on the role of the arbitrageur is still valid, the extent to which
the arbitrageur alleviates the investors’ constraints may be smaller than in the
competitive case.
Finally, we consider a case in which the (competitive) arbitrageur is endowed with
capital and is subject to margin requirements that limit the size of his position in
proportion to the total amount of his capital. His capital is owned and traded by the
investors. This variation appears more realistic than our basic model, and is also richer in
implications. Because the investors may now invest in arbitrage capital, in equilibrium thereturn on arbitrage must be consistent with other investment opportunities; this allows us
to explicitly solve for the unique amount of arbitrage capital that is consistent with
equilibrium. Our approach is consistent with recent empirical work on the profitability of
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arbitrage activity (e.g., Mitchell et al., 2002), which suggests that market imperfections
severely limit the return on arbitrage, and may drive it to a level that does not dominate
other investment opportunities.
The setup of this paper builds on the work of Detemple and Murthy (1994, 1997), who
solve for equilibrium in the presence of heterogeneous agents, both with and withoutportfolio constraints, but not with redundant derivative securities. Our framework more
closely follows the production economy of Detemple and Murthy (1994), in which the
technology determines the stock price dynamics exogenously and much of the action is
picked up by the bond interest rate. This, along with logarithmic investor preferences,
leads to considerable tractability and closed-form solutions in our analysis. Basak and
Croitoru (2000) go one step further by endogenizing the arbitrage opportunity and show
that in the presence of heterogeneous constrained investors such as our own, the mispricing
and arbitrage opportunity will be present under a broad range of circumstances. None of
these papers, however, includes a specialized arbitrageur.
To our knowledge, little work exists that examines the impact of specialized arbitrage in
an equilibrium in which all agents are rational. Related to our work is Gromb and
Vayanos (2002). These authors’ goals are somewhat different from ours, however, in that
they focus on the issue of a competitive arbitrageur’s welfare impact. In order to do this,
they examine a different form of constraints, specifically, that of a segmented market for
the risky assets, where different investors hold different risky assets. Without the
arbitrageur there is no trade; in the presence of the arbitrageur, trade occurs and a Pareto
improvement takes place (albeit not necessarily in a Pareto-optimal way). This is consistent
with the arbitrageur’s role in our model, which allows more trade to take place between
investors. We would not expect to obtain such a clear welfare result, however, since in ourmodel there is trade even in the absence of an arbitrageur and thus our benchmark is much
more complex. Despite improved risk-sharing, the overall welfare effect of our arbitrageur
is ambiguous, because his presence may move prices in a way that can be unfavorable to
the investors (Section 4). Our model is also closer in spirit to the standard asset pricing
models (and thus easier to relate to observable economic variables or well-understood
benchmarks). Loewenstein and Willard (2000) provide an equilibrium in which an
arbitrageur (interpreted as a hedge fund) plays a similar role when investors are subject not
to portfolio constraints, but to uncertain timing for their consumption. Unlike in our
model, arbitrage trades in Loewenstein and Willard are long lived.
Somewhat related are works, including Brennan and Schwartz (1990) and Liu andLongstaff (2004), that study a constrained arbitrageur’s optimal policy given the arbitrage
opportunity is present. Liu and Longstaff demonstrate that an arbitrageur may optimally
underinvest in an arbitrage opportunity, known to vanish at some future time, by taking a
smaller position than margin requirements would allow. Finally, our analysis complements
an important and growing strand of the finance literature that is often known as the study
of ‘‘limits on arbitrage.’’ This approach, inaugurated by De Long et al. (1990), is
thoroughly surveyed in Shleifer (2000). In these papers, arbitrage opportunities are
generated by the presence of irrational noise traders, and are allowed to subsist in
equilibrium due to various market imperfections (e.g., portfolio constraints, transaction
costs, short-termism, and model risk). Papers in this area that are particularly related toour work include Xiong (2001), who studies the impact of arbitrageurs (‘‘convergence
traders’’) on volatility. Kyle and Xiong (2001) extend his model to the study of contagion
effects between the markets for two risky assets, only one of which is subject to noise trader
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risk. Attari and Mello (2002) focus on the effect of trading constraints on the arbitrageur’s
impact and conclude (as we do) that these may have radical effects. Unlike the majority of
the literature, they assume that arbitrageurs take into account how their own trades affect
prices, as we do in part of our analysis.
Our work also complements the growing literature on asset prices in the presence of heterogeneous beliefs and short-sale restrictions, including Harrison and Kreps (1978),
Detemple and Murthy (1997), and Scheinkman and Xiong (2003). In these models, an
asset price may exceed its fundamental value (its valuation based on investors’ own
marginal rates of substitution) by a quantity reflecting the investors’ anticipated
speculative gains. There, investors are willing to pay more than the fundamental value
of an asset since ownership of the asset gives them the right to resell the asset to another
investor at an even higher price in the future. Some of these models’ features are also
present in our economy, which contains a short-sale constrained stock and heterogeneous
beliefs (but also a derivative security and an arbitrageur, not present in these papers). In
the presence of the redundant derivative and investment restrictions, our setting
additionally leads to riskless arbitrage opportunities, enabling a study of the role of an
arbitrageur—our primary focus.
The rest of the paper is organized as follows. Section 2 provides our single-period, finite-
state example on the role of the arbitrageur. Section 3 introduces our continuous-time
setup, and Section 4 describes the equilibrium with a competitive arbitrageur. Sections 5
and 6 are devoted to extensions to the case of a noncompetitive arbitrageur, and
to an arbitrageur endowed with capital and subject to margin requirements, respectively.
Section 7 concludes, and the Appendix A provides all proofs.
2. The role of the arbitrageur: an example
To convey our main intuition on the role of the arbitrageur and show that our main
point does not depend on the modeling setup employed, we start with an example in a
simple, one-period framework. Portfolio decisions are made at time 0 and securities pay off
at time 1. We first consider an economy with two heterogeneous investors i ¼ 1; 2 and no
arbitrageur. Consider two zero-net-supply securities, S and P , with time-0 prices also
denoted by S and P , and with identical payoffs of one unit if state o occurs, and zero
otherwise. The only difference between S and P is how trading in each is constrained.
Denoting by ai j investor i ’s investment, measured in number of shares, in security j (where j ¼ S ; P ), we assume that both investors’ holdings are subject to the position limits
ai S pb; ai
P XÀ g; i ¼ 1; 2. (1)
For simplicity, we assume that S and P are the only two available securities paying off in
state o. Then, investor i ’s payoff if state o occurs is given by Ai ¼ ai S þ ai
P . The constraints
in (1) alone do not place any restrictions on the choice of Ai , because one could always use
S to take on unlimited short positions and P to take on unlimited long positions.
Nonetheless, only a limited set of payoffs are available to the investors in equilibrium: No
investor can go long more than g shares of P , because the other investor cannot act as
a counterparty without violating his constraint on short-sales of P . Hence, no investor canreceive a state-o payoff greater than g þ b. A similar argument shows that no investor can
take on an unlimited short position, because the other investor cannot provide the
necessary counterparty. Thus, in equilibrium, both investors’ state-o payoffs must satisfy
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the constraint
Àb À gpAi pb þ g; i ¼ 1; 2. (2)
In short, investors’ position limits prevent them from acting as a counterparty to each
other, and this limits the trades that are possible in equilibrium.We now examine how the feasible trades (i.e., such that (2) holds) are affected in the
presence of a third agent, an arbitrageur. The arbitrageur is assumed to only take on
riskless arbitrage positions to take advantage of any price difference between S and P .
Whenever P aS , securities that have identical payoffs trade at different prices, giving rise
to a riskless arbitrage opportunity. For example, when P 4S , an arbitrage position
consisting of going long one share of S and short one share of P provides a time-0 profit
equal to P À S 40 without any future cash flows, and hence no risk. Basak and Croitoru
(2000) show how such price discrepancies arise in equilibrium in the presence of
constrained heterogeneous investors. In our subsequent analysis, we provide equilibria in
which such mispricings also arise endogenously. For now, we take the presence of
mispricing ðP 4S Þ as given and address the issue of the economic role of an arbitrageur
who exploits the mispricing. It is straightforward to verify that, given the constraints (1),
the other direction of mispricing ðS 4P Þ cannot arise in equilibrium, as agents would then
add an unbounded arbitrage position to their portfolio. Furthermore, we take the size of
the arbitrageur’s position (a3S 40 shares of S , a3
P ¼ Àa3S shares of P ) as given. In the
absence of any constraint, an arbitrageur would optimally choose to hold an infinitely
large arbitrage position and thus equilibrium would be impossible. To avoid this, later
sections assume either that the arbitrageur is constrained or that he is non-price-taking.
Then, an investor who desires a large short position can short as many as b þ a3
S sharesof S (instead of only b if the arbitrageur were not present), because the arbitrageur acts as
an additional counterparty over and above the other investor. A similar argument applies
to large long positions, so that the constraint on investors’ state-o payoffs (the analogue of
(2)) is now
Àb À g À a3S pAi
pb þ g þ a3S ; i ¼ 1; 2. (3)
Eq. (3) reveals that the presence of the arbitrageur makes it possible for investors to trade
more. To see the intuition for this, take the example of investor 1 desiring to buy a large
payoff from investor 2 (possibly because the former is more optimistic, or less risk averse).
Without an arbitrageur, once investor 1 has purchased b shares of S and g shares of P ,trading between the two investors becomes ‘‘stuck’’ no matter how great the heterogeneity
between them. The arbitrageur makes it possible for investor 1 to buy an extra a3S shares of
P from him. This position is converted by the arbitrageur into a position in S , that is then
voluntarily shorted by investor 2, given the constraints were preventing him from shorting
more before the arrival of the arbitrageur. The net result is that the arbitrageur has made it
possible for the two investors to trade more risk with each other, even though he does not
take on any risk himself. The arbitrageur plays the role of a financial intermediary: The
larger his position, the more he alleviates the constraints.
It should be pointed that the role of the arbitrageur is specific. Adding a third investor
(who can take on risk), instead of an arbitrageur, to the economy can either worsen orrelieve the constraint in (2). In the presence of mispricing, the third investor will always
bind on one of his position limits (1); which one he binds on depends on his desired cash
flow. If this third investor (indexed by 3Ã) binds on his upper constraint on S , the
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constraint on investor i 2 1; 2’s state-o cash flow becomes
À2b À gpAi pb þ g À a3Ã
P . (4)
While the lower bound is lower than in (3), showing an alleviation of the lower constraint,
the upper bound can be either lower or higher, and accordingly the constraint either moreor less stringent. A similar point can be made when agent 3 Ã binds on his constraint on P .
The effect of a third investor is thus ambiguous, highlighting the specific role of an
arbitrageur, who always improves risk-sharing.
While this section deals with a simplistic economic setup, in Section 4 we show how the
exact same points can be made within a much more general continuous-time modeling
framework. It should also be noted that the analysis in this section is robust to changes in a
large number of assumptions: The preferences, beliefs, and endowments of investors 1 and
2, the type of portfolio constraints (position limits or constraints on portfolio weights,
whether they are one- or two-sided, whether they are homogeneous or heterogeneous
across agents), and the net supplies of the securities (meaning that our analysis is also
equally valid in production and exchange economies). All that is really needed is the
presence of redundant securities, portfolio constraints, and heterogeneity amongst
investors.
3. The economic setting
This section describes the basic economic setting, which is a variation on the Cox,
Ingersoll and Ross (1985) production economy. The economy is populated by two
investors, who trade so as to maximize the cumulated expected utility of consumption in astandard fashion, and an arbitrageur, who is constrained to hold only riskless arbitrage
positions. The description of the arbitrageur is relegated to later sections, as related
assumptions will be different in each section.
3.1. Investment opportunities
We consider a finite-horizon production economy. The production opportunities are
represented by a single stochastic linear technology whose only input is the consumption
good (the numeraire). The instantaneous return on the technology is given by
dS ðtÞ
S ðtÞ¼ mS dt þ sdwðtÞ, (5)
where mS and s ðs40Þ represent the constant mean return and volatility on the technology,
respectively, S is the amount of the good invested, and w is a one-dimensional Brownian
motion. For convenience, we often refer to the technology as the ‘‘stock’’ S .
The two investors observe the return of the stock, but have incomplete (though
symmetric) information on its dynamics. They know s (from the stock return’s quadratic
variation), but must estimate mS via its conditional expectation and rationally update their
beliefs in a Bayesian fashion with heterogeneous prior beliefs. We denote by mi S ðtÞ the
conditional estimate of mS by investor i ¼ 1; 2 at time t, given his prior beliefs and hisobservation of the stock’s realized return; by ¯ mðtÞ ¼ ðm1
S ðtÞ À m2S ðtÞÞ=s we denote
the normalized difference between the two investors’ estimates. The process ¯ m represents
the disagreement between the investors on their relative optimism or pessimism about the
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stock mean return. The process ¯ m is given exogenously, since all learning about mS comes
from observing the (exogenous) realized return on the stock. For simplicity, we shall
assume that ¯ m40, and we call investor 1 (2) the optimistic (pessimistic) investor. Consider
the well-known Gaussian example in which investor i ’s prior belief is normally distributed
with mean mi ð0Þ and variance vð0Þ. Then, investors’ estimates have dynamics dmi S ðtÞ ¼
ðvðtÞ=sÞ dwi ðtÞ, where vðtÞ ¼ vð0Þs2=ðvð0Þt þ s2Þ, implying d ¯ mðtÞ ¼ ÀðvðtÞ=sÞ ¯ mðtÞ dt, or ¯ mðtÞ ¼
¯ mð0Þ½s2=ðvð0Þt þ s2Þs. Assuming ¯ mð0Þ40 implies ¯ mðtÞ40, 8t. Our subsequent analysis goes
through for ¯ mo0, with the only (minor) difference being the presence of another mispricing
case arising in equilibrium (Sections 4–6), that mirrors the mispricing case arising for ¯ m40.
Finally, we let
wi ðtÞ ¼1
s
Z t
0
dS ðsÞ
S ðsÞÀ
Z t
0
mi S ðsÞ ds
!(6)
denote investor i ’s estimate of the Brownian motion w. By Girsanov’s theorem, w
i
is astandard Brownian motion under investor i ’s beliefs, and investor i ’s perceived stock
return dynamics are given by
dS ðtÞ
S ðtÞ¼ mi
S ðtÞ dt þ sdwi ðtÞ; i ¼ 1; 2. (7)
Though not a central feature of our model, heterogeneity in beliefs is employed to generate
trading activity between the logarithmic investors.
In addition to the stock, investors may invest in two zero-net-supply, non-dividend-
paying securities. One is an instantaneous riskless bond with return
dB ðtÞ
B ðtÞ¼ rðtÞ dt, (8)
and the other a risky derivative with perceived return process
dP ðtÞ
P ðtÞ¼ mi
P ðtÞ dt þ sdwi ðtÞ; i ¼ 1; 2. (9)
Similarly to the stock, investors observe the derivative return process but not its dynamics
coefficients, and hence they draw their own inferences about the derivative mean return mP .
Since the derivative pays no dividends, we take its volatility parameter s (assumed to equal
stock volatility for simplicity) to define the derivative contract P , as is standard in theliterature (e.g., Karatzas and Shreve, 1998). The interest rate r and derivative mean return
mi P , on the other hand, are determined endogenously in equilibrium. Any zero-net-supply
security is an example of this derivative. Real-life examples that are particularly close in
spirit to our idealized derivative include a stock index futures contract, or a total return
swap on a stock index; we take the latter as a leading example in our subsequent
discussion. In the case of the derivative being an overnight total return swap on the stock, a
long investor receives the total stock return and pays an instantaneously riskless swap rate
rP (possibly different from r), with each unit yielding between t and t þ dt:
dS ðtÞS ðtÞ
À rP ðtÞ dt ¼ ½mi S ðtÞ À rP ðtÞ dt þ sdwi ðtÞ; i ¼ 1; 2. (10)
All of our results, derived for the general case of a derivative with mean return mi P , are
equally valid in the case of the total return swap if we take rP ðtÞ ¼ rðtÞ þ mi S ðtÞ À mi
P ðtÞ. The
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generalization to the case of the derivative having a different volatility than the stock is
straightforward and is considered in an earlier version of this manuscript (Basak and
Croitoru, 2003). Apart from increased notational complexity, all our main results here are
robust to such a generalization. Moreover, defining the derivative by a dividend process
would not entail major changes to our analysis. All of the expressions in the paper wouldremain valid but the derivative volatility s would be endogenously determined via a
present value formula. Our main points and intuition would be unaffected.
3.2. Investors’ optimization problems
With all the uncertainty generated by a one-dimensional Brownian motion, the stock
and derivative returns are perfectly correlated, and so the derivative is redundant.
However, restrictions on investors’ risky investments generate an economic role for the
derivative. Letting yi ðyi
B ;yi
S ;yi
P Þ, with yi
B ðtÞ ¼ W i ðtÞ À yi
S ðtÞ À yi
P ðtÞ, denote the amounts
of investor i ’s wealth, W i , invested in B , S , and P , respectively, we assume that in addition
to short sales being prohibited, stock investments face leverage limits, and investments in
the derivative have limited short-selling restrictions at all times:
0pyi S ðtÞpbW i ðtÞ; yi
P ðtÞXÀ gW i ðtÞ, (11)
where bX1, g40. For retail investors, the stock investment restriction may be due to
margin requirements, while for mutual funds they may represent leverage restrictions. If
the derivative security is an over-the-counter contract, such as a total return swap, the
derivative investment constraint (11) may be due to a collateral requirement imposed by a
counterparty. If the derivative is an exchange-traded contract, such as a futures contract,then the restriction (11) may represent margin regulations imposed by the exchange. The
one-sided restriction on holdings of the derivative is for expositional simplicity, with
the two-sided case being a straightforward extension that does not affect our main
conclusions. In particular, suppose that the derivative holdings are additionally restricted
to satisfy the leverage restriction yi P ðtÞpgW i ðtÞ, g40. Then, there would be additional
scenarios in the investors’ optimal holdings of Proposition 3.1, leading to additional
equilibrium cases as discussed in Section 4.2.
The risky assets are exposed to the same amount of risk ðsdwi ðtÞÞ, and thus it is natural
to expect them to be priced accordingly and to offer identical mean returns. In fact, if this
were not the case, there would be an arbitrage opportunity available, exploited by short-selling the asset with the lower mean return and investing the proceeds in the other, which
would generate a riskless arbitrage profit proportional to the difference in mean returns.
This motivates our definition of mispricing between the stock and derivative under investor
i ’s beliefs as the difference between their mean returns,
Di S ;P ðtÞ mi
S ðtÞ À mi P ðtÞ. (12)
We note that the observed return agreement across investors enforces agreement on the
mispricing: D1S ;P ðtÞ ¼ D2
S ;P ðtÞ DS ;P ðtÞ. In the presence of investment restrictions, DS ;P need
not be zero. If DS ;P ðtÞ40, the stock has a higher expected return and we say that it is
‘‘cheap’’ relative to the derivative; conversely, if DS ;P ðtÞo0, the stock is more ‘‘expensive’’(and the derivative is cheaper). For example, if the derivative is the overnight total return
swap, the mispricing is given by any deviation of the swap rate from the interest rate since,
whenever rP ar, an arbitrage opportunity is available. If rP 4r, the swap is more expensive
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than the stock, and a strategy consisting of buying the stock, financing the purchase by
riskless borrowing, and taking a short position in the swap yields an instantaneous
arbitrage profit of $ðrP À rÞ per dollar in the stock; vice versa if rP or.
Investor i is endowed with initial wealth W i ð0Þ. He chooses consumption policy ci and
investment strategy yi so as to maximize his expected lifetime logarithmic utility subject toa dynamic budget constraint and position limits, that is, to solve the problem
maxci ; yi
E i Z T
0
logðci ðtÞÞ dt
!(13)
s. t. dW i ðtÞ ¼ ½W i ðtÞrðtÞ À ci ðtÞ dt þ fyi S ðtÞ½mi
S ðtÞ À rðtÞ þ yi P ðtÞ½mi
P ðtÞ À rðtÞg dt
þ ½yi S ðtÞ þ yi
P ðtÞsdwi ðtÞ ð14Þ
and ð11Þ.
This differs from the standard frictionless investor’s dynamic optimization problem
due to the investment restrictions, potential redundancy, and mispricing between the
risky investment opportunities S and P . The solution of investor i ’s problem is reported
in Proposition 3.1. Cases in which P is cheap are ignored, as they cannot arise in
equilibrium.
Proposition 3.1. Investor i ’s optimal consumption and investments in the stock S and
derivative P are given by
c
i
ðtÞ ¼
W i ðtÞ
T À t ; (15)
when there is no mispricing, i.e. DS ;P ðtÞ ¼ 0,
yi S ðtÞ
W i ðtÞ¼
2 ½0;b
0;
&yi
P ðtÞ
W i ðtÞ¼
XÀ gs; s.t.yi
S ðtÞ
W i ðtÞþ
yi P ðtÞ
W i ðtÞ¼mi
S ðtÞ À rðtÞ
s2(a)
if mi S ðtÞXrðtÞ À gs2
Àg if mi S ðtÞorðtÞ À gs2 (b)
8>>>><>>>>:
(16)
and when there is mispricing, with the stock being cheaper than the derivative, i.e. DS ;P ðtÞ4
0,
yi S ðtÞ
W i ðtÞ¼
b if mi S ðtÞ4mi
P ðtÞXrðtÞ À ðg À bÞs2
b if mi S ðtÞ4rðtÞ À ðg À bÞs2
4mi P ðtÞ
mi S ðtÞ À rðtÞ
s2þ g if rðtÞ À ðg À bÞs2Xmi
S ðtÞXrðtÞ À gs2
0 if rðtÞ À gs24mi
S ðtÞ;
8>>>>>><>>>>>>:
(c)
(d)
(e)
(f)
(17)
yi P ðtÞW i ðtÞ
¼
mi P ðtÞ À rðtÞ
s2À b if (c)
Àg if (d)
Àg if (e)
Àg if (f):
8>>>>><>>>>>:
(18)
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An investor can be in one of six possible scenarios, (a)–(f), depending on the economic
environment and whether there exists mispricing (cases (c)–(f)) or not (cases (a)–(b)). When
there is no mispricing and the perceived mean return is sufficiently low, the investor desires
the lowest possible exposure to risk and thus binds on his lower constraint on both risky
assets (case (b)). Otherwise, he is indifferent between all feasible investments in the tworedundant risky assets that lead to his optimal risk exposure (case (a)) (that is, investments
ensure that the volatility of his portfolio equals the market price of risk).
Under mispricing (cases (c)–(f)), a riskless, costless, profitable arbitrage opportunity is
available, and the investor indulges in it to the greatest extent possible as allowed by the
position limits. Effectively, the investor adds to his portfolio an arbitrage position
consisting of a long position in the cheap stock and a short position in the expensive
derivative. This implies that the investor is always binding on at least one of the investment
restrictions in (11), and thus uniquely determines the allocation between the stock and the
derivative. When the investor is relatively optimistic about the mean return of the stock
and the derivative is not too expensive (case (c)), he desires a high risk exposure, wishing to
go long in both risky assets, and so ends up binding on the leverage constraint on S . When
the derivative is more expensive (DS ;P 40 or, in the case of a total return swap, higher rP )
by a large amount, he is driven to his short sale limit on P ; if additionally the investor is
pessimistic about the stock, he desires a low risk exposure and thus does not invest in the
stock (case (f)), otherwise he goes long in the stock (cases (d)–(e)). As is demonstrated in
Section 4.2, mispricing cases (c) and (e) may occur in equilibrium (with or without an
arbitrageur). Cases (d) and (f) do not.
4. Equilibrium with a competitive arbitrageur
4.1. The arbitrageur
We assume that in addition to the two investors i ¼ 1; 2, the economy is populated by an
arbitrageur indexed by i ¼ 3. The arbitrageur is assumed to take on only riskless, costless
arbitrage positions, and to maximize his expected cumulated profits under a limit on the
size of his positions. This natural formulation of the arbitrageur is somewhat simplistic,
but it does capture the most important distinguishing characteristics of a specialized
arbitrageur such as an arbitrage desk in an investment bank or a hedge fund focusing on
arbitrage strategies. Denoting the arbitrageur’s (dollar) investments by y3
ðy3B ; y
3S ; y
3P Þ,
his problem can be expressed as:
maxy3
E 3Z T
0
C3ðtÞ dt
!(19)
s.t. C3ðtÞ dt ¼ y3S ðtÞm3
S ðtÞ þ y3P ðtÞm3
P ðtÞ þ y3B ðtÞrðtÞ
È Édt
þ ½y3S ðtÞ þ y3
P ðtÞsdw3ðtÞ, ð20Þ
y3S ðtÞ þ y3
P ðtÞ þ y3B ðtÞ ¼ 0, (21)
ðy3S ðtÞ þ y3
P ðtÞÞs ¼ 0, (22)
0py3S ðtÞpM . (23)
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An equivalent interpretation of the no-risk condition (22) is that the arbitrageur’s position
is uncorrelated with the stock return, as would ideally be the case for a hedge fund’s return.
The arbitrageur may be a trader of an investment bank or a hedge fund. For an investment
bank, the investment restriction (23) imposed on the arbitrage trader may be linked to the
capital requirements faced by the bank, with a credit limit controlled by the FederalReserve. In the case of a hedge fund, the investment restriction imposed on the arbitrageur
may simply represent an internal position limit.
Conditions (21)–(22) imply
y3P ðtÞ ¼ Ày3
S ðtÞ; y3B ðtÞ ¼ 0, (24)
so it follows that the arbitrageur’s holdings in all securities are pinned down by his stock
investment. Therefore, the constraint on holdings of the stock (23) also limits the
arbitrageur’s holdings of the other securities.
In the absence of mispricing ðDS ;P ¼ 0Þ, it is straightforward to verify that the
arbitrageur is indifferent between all portfolio holdings that satisfy (23)–(24), since they all
provide him with zero cash flows (and he can do no better than this). In the presence of
mispricing ðDS ;P 40Þ, however, as long as y3S 40, the arbitrageur receives a profit that is
proportional to y3S . The optimal policy then is to choose the maximal possible value for y3
S ,
namely M . From (24), the other holdings are given by y3P ðtÞ ¼ ÀM and y3
B ðtÞ ¼ 0, and the
arbitrageur’s instantaneous profit rate is given by
C3ðtÞ ¼ y3S ðtÞm3
S ðtÞ þ y3P ðtÞm3
P ðtÞ ¼ M DS ;P ðtÞ. (25)
For example, in the case of the derivative being a total return swap on the stock, the
arbitrageur may be purchasing the stock, financing the purchase by borrowing at theriskless rate and simultaneously entering the swap, whereby he pays the stock return and
receives the swap rate rP . In the absence of mispricing, the two interest rates are equal.
Under mispricing, the notional swap rate is higher than the interest rate (by DS ;P ), and the
difference generates the arbitrageur’s profit.
It should be pointed out that the price-taking arbitrageur’s solution, as well as the value
of his profits, is independent of his beliefs and preferences. Due to his constraints,
(21)–(22), the arbitrageur consumes his profits as they are made ðc3 ¼ C3Þ and holds no net
wealth, so that without loss of generality, he can be assumed to be risk neutral, without any
time preference. Any increasing utility function for the arbitrageur would lead to the same
behavior as for the risk-neutral arbitrageur, in this section as well as in Sections 5 and 6.
4.2. Analysis of equilibrium
We now proceed with the description of the equilibrium (defined, in standard fashion, as
the security prices such that agents’ optimal policies clear the security markets, given the
prices) in our economy. For convenience, quantities of interest are expressed as a function
of the aggregate wealth, W W 1 þ W 2, and the proportion of aggregate wealth held by
investor 1, l W 1=W . Several cases are possible in equilibrium, depending on investor 1
and 2’s binding constraints. We denote each case by the optimization case of each investor;
for example, in equilibrium ða; aÞ each investor is in case (a), in equilibrium ða; bÞ investor 1is in case (a) and investor 2 in case (b), etc.
Proposition 4.1 reports the possible equilibrium cases, details the conditions for their
occurrence, and characterizes economic quantities (prices and consumptions) in each case.
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Proposition 4.1. In equilibrium, the investors’ optimization cases, equilibrium mispricing and
interest rate, and distribution of wealth dynamics are as follows.
When
¯ mðtÞp gslðtÞ
þ min slðtÞ
; s1 À lðtÞ
ðb À 1Þ þ M lðtÞW ðtÞ
!& ', (26)
investors are in ða; aÞ and
DS ;P ðtÞ ¼ 0; rðtÞ ¼ lðtÞm1S ðtÞ þ ð1 À lðtÞÞm2
S ðtÞ À s2, (27)
and
dlðtÞ ¼ lðtÞð1 À lðtÞÞ2ð ¯ mðtÞÞ2 dt þ lðtÞð1 À lðtÞÞ ¯ mðtÞ dw1ðtÞ. (28)
When
¯ mðtÞ4 ðg þ 1ÞslðtÞand lðtÞX 1b 1 À M W ðtÞ
, (29)
investors are in ða; bÞ and
DS ;P ðtÞ ¼ 0; rðtÞ ¼ m1S ðtÞ À
g þ 1
lðtÞÀ g
!s2, (30)
and
dlðtÞ ¼ sðg þ 1Þð1 À lðtÞÞ2
lðtÞdt þ ð1 À lðtÞÞ dw1ðtÞ
!. (31)
When
¯ mðtÞ4gs
lðtÞþ
s
1 À lðtÞb À 1 þ
M
lðtÞW ðtÞ
and lðtÞo
1
b1 À
M
W ðtÞ
, (32)
investors are in ðc; eÞ and
DS ;P ðtÞ ¼ ¯ mðtÞs Àðb À 1Þs2
1 À lðtÞÀgs2
lðtÞÀ
M s2
lðtÞð1 À lðtÞÞW ðtÞ40, (33)
rðtÞ ¼ m2S ðtÞ À ð1 À gÞs2 þ ðb À 1Þ
lðtÞ
1 À lðtÞ
þM
ð1 À lðtÞÞW ðtÞ !s2, (34)
and
dlðtÞ ¼ lðtÞ DS ;P ðtÞ b À 1 þM
W ðtÞ
!þ ð1 À lðtÞÞð ¯ mðtÞ À DS ;P ðtÞ=sÞÂ Ã2
& 'dt
þ lðtÞð1 À lðtÞÞð ¯ mðtÞ À DS ;P ðtÞ=sÞ dw1ðtÞ. ð35Þ
In all cases, the aggregate wealth dynamics follow:
dW ðtÞ ¼ W ðtÞ m1S ðtÞ À
1
T À t
À M DS ;P ðtÞ
!dt þ W ðtÞsdw1ðtÞ (36)
and investor i ’s consumption and the arbitrageur’s profit are given by
c1ðtÞ ¼lðtÞW ðtÞ
T À t; c2ðtÞ ¼
ð1 À lðtÞÞW ðtÞ
T À tand C3ðtÞ ¼ M DS ;P ðtÞ. (37)
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Three cases are possible in equilibrium: ða; aÞ and ða; bÞ in which there is no mispricing
and the arbitrageur makes no profits, and ðc; eÞ in which mispricing occurs. We now
describe how, as the divergence in beliefs across investors grows, the equilibrium may move
into region ðc; eÞ, where the arbitrageur is active and affects the equilibrium.
In our economy, investors trade due to their disagreement on the mean stock return,with the optimist buying stock from the pessimist. If the disagreement is not too large
(equilibrium case ða; aÞ), then the optimist puts more than 100% of his wealth into stock
and the pessimist puts some fraction between 0% and 100% into stock. Since the
technology return determines the stock price, to clear the market, the interest rate is set at a
value depending on the average degree of optimism across investors (as in the framework
of Detemple and Murthy, 1994). As the optimist becomes even more optimistic, he buys
more and more stock from the pessimist, until either the optimist hits his leverage
constraint or the pessimist hits his no-short-sale constraint. Which constraint is hit first
depends on the distribution of wealth between the two investors, with the poorer one
having a tendency to hit his constraint first. Let us suppose that the pessimist hits his
constraint first. Then, as the optimist becomes even more optimistic beyond this point, he
buys derivatives from the pessimist, even though the pessimist has more stock that he could
sell to the optimist if the optimist were not leverage constrained. As the optimist becomes
even more optimistic, he eventually buys so many derivatives from the pessimist that the
pessimist hits his constraint on short derivative positions.
At this point, additional trades between the optimist and the pessimist would be
impossible, and risk-sharing would be ‘‘stuck,’’ were it not for the presence of the
arbitrageur. As the optimist becomes even more optimistic, the arbitrageur buys stock
from the pessimist (who still has some stock left because he did not hit his no-short-salesconstraint before the optimist hit his leverage constraint) and sells derivatives to the
optimist. The arbitrageur is essentially converting stock positions into derivative positions.
As the optimist continues to become more optimistic, he buys more and more derivative,
until a point is reached at which the arbitrageur hits his position limit M or the pessimist
sells the arbitrageur all of his stock. Let us assume that the arbitrageur’s position limit is
hit first, which occurs when the economy moves from region ða; aÞ into region ðc; eÞ, and the
derivative becomes mispriced relative to the stock. As the optimist becomes more
optimistic beyond this point, the arbitrageur cannot sell him more derivative. Instead, the
derivative’s mean return is reduced and the optimist limits the amount of derivative that he
buys because it becomes more expensive.Returning to the case of the derivative being a total return swap, a positive mispricing
means that the notional riskless rate used to price total return swaps becomes higher than
the actual riskless rate in the economy. This is the opposite of what happens when there is a
‘‘repo squeeze,’’ and the notional swap rate is less than the market interest rate. This higher
notional interest rate makes the total return swaps more costly to the optimist, and entices
him to reduce his holding to the level the arbitrageur is allowed by his position limit to sell
to the optimist. On the other side of the book, however, the arbitrageur has a plentiful
supply of stock for sale by the pessimistic investor. Thus, for markets to clear, the riskless
rate needs to drop below the swap rate; this induces the pessimist to reduce the amount of
stock he wants to sell to the amount the arbitrageur can buy (M ).If, on the other hand, the pessimistic investor is much wealthier than the optimist, as
divergence in beliefs grows the optimist hits his leverage constraint on the stock before the
pessimist hits his constraint on the derivative. If this is the case, market clearing does not
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require that the optimist’s demand of the derivative be reduced, because the pessimist is
wealthy enough to act as a counterparty and sell to the optimist derivatives without hitting
his constraint; thus, there is no mispricing in equilibrium (case ða; bÞ).
In short, the arbitrageur is active under high divergence in beliefs across investors and
provided the pessimist is not much wealthier than the optimist. The arbitrageur buys stockfrom the pessimist and sells derivatives to the optimist. Under mispricing (equilibrium case
ðc; eÞ), the derivative is more expensive than the stock, and the arbitrageur profits from the
difference. If we also assume that the investors face a leverage constraint on investments in
the derivative ðyi P ðtÞpgW i ðtÞ; g40Þ, an additional case may occur in equilibrium, in which
there is no arbitrage opportunity, and hence no role for the arbitrageur. In that case, the
leverage constraint prevents the optimist from purchasing more derivatives than what
the pessimist can short, and so the equilibrium does not require the derivative to become
less attractive to the optimist via the mispricing. The optimist is prevented by the
additional leverage constraint from purchasing derivatives from the arbitrageur, who in
turn cannot trade. Our analysis of region ðc; eÞ remains unchanged, however, and our
main message on the role of the arbitrageur is thus unaffected by this generalization of our
basic setup.
Henceforth we focus on the analysis of region ðc; eÞ, where the arbitrageur has an
economic impact and so affects economic quantities. In region ða; aÞ, the equilibrium is as
in an unconstrained economy as analyzed by Detemple and Murthy (1994), because no
constraints are binding; in region ða; bÞ, the equilibrium is similar to a constrained
economy without a derivative as in Detemple and Murthy (1997). In both of these cases,
there is no arbitrage opportunity available. It is straightforward to verify that region ðc; eÞ
does occur for plausible parameter values: For example, according to (32), if g ¼ 0:5,b ¼ 1:5, s ¼ 0:1, M =W ðtÞ ¼ 0:1, and lðtÞ ¼ 0:5, region ðc; eÞ occurs whenever investor
1’s estimate of the mean return of the stock exceeds investor 2’s estimate by at least
2.4% per year. In our analysis, we focus on the terms reflecting the effects of the
arbitrageur’s presence and the size of his position (measured by M ). All of our analyses
and comparisons are valid only for given aggregate wealth ðW Þ and distribution of wealth
across investors ðlÞ. For comparison with our economy, we introduce the following two
benchmarks:
Economy I: no constraints, no derivative; otherwise, as our economy.
Economy II: no arbitrageur; otherwise, as our economy.
In Economy I, all equilibrium quantities are as in our region ða; aÞ, while in Economy II,they are as in our economy for the particular case M ¼ 0.
The value of the mispricing can be interpreted as the amount of heterogeneity in beliefs
that investors cannot trade on due to the binding constraints. Whenever no constraints are
binding and thus risk-sharing is not limited, the mispricing is always equal to zero.
However, on moving from region ða; aÞ to region ðc; eÞ, the mispricing starts at a value of
zero and then increases linearly with heterogeneity ð ¯ mÞ while interagent transfers remain
stuck due to the constraints. The mispricing is decreased by the presence of the arbitrageur,
and decreases further as the size of his position increases; compared with our benchmarks,
we have DIS ;P ¼ 0oDS ;P oD
IIS ;P . Keeping in mind our interpretation of the mispricing as the
amount of untraded heterogeneity, this suggests an improvement in risk-sharing due to thearbitrageur’s presence.
The value of the interest rate is increased by the presence of the arbitrageur; we have
rI4r4rII. This is indirect evidence of an improvement in risk-sharing between investors
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due to the arbitrageur’s presence: improved risk-sharing decreases investors’ precautionary
savings. For the bond market to clear in spite of this, it is necessary for the interest rate to
rise, thus becoming closer to its value in an unconstrained economy. Note that, as in the
production economy of Detemple and Murthy (1994), since the technology determines the
stock price dynamics exogenously, it is the interest rate that has to adjust for markets toclear and thus it picks up much of the action. The interest rate can be solved for explicitly
because investor demands are myopic and linear in the stock’s Sharpe ratio (i.e., entirely
determined by the interest rate).
In short, the impact of the arbitrageur on all price dynamics parameters is to make them
closer to their values in an unconstrained economy; the larger his position, the closer the
resulting equilibrium is to an unconstrained equilibrium. The conditions for region ðc; eÞ to
occur (and the constraints to bind) are also affected. From (32), it is clear that the larger
the arbitrageur’s position, the less likely it is for region ðc; eÞ to occur and the constraints to
bind, perturbing risk-sharing between investors. In all cases, the effect of increasing the size
of the arbitrageur’s position (M ) on the expressions is tantamount to an increase in b or g
(i.e., an alleviation of the constraints). All of this suggests that the arbitrageur alleviates the
effect of the portfolio constraints. The overall effect of his presence on investor welfare is
ambiguous, however, because his trades move prices in a way that can hurt the investors.
The arbitrageur’s trades reduce the mean stock excess return (above the riskless rate),
which hurts both investors, but increase the mean derivative return, benefiting the
optimistic investor (who is long) and hurting the pessimist. The increase in the interest rate
benefits the pessimist (long in the bond), but hurts the optimist. Thus, in spite of the
unambiguous improvement in risk-sharing due to the arbitrageur’s trades, the welfare
impact of the arbitrageur’s presence is ambiguous. However, note that all agents in ourmodel are better off than in a no-trade economy, as in Gromb and Vayanos (2002),
because they could choose to not trade.
Finally, interpreting the leverage limit parameter b and the limited short-selling
parameter g as policy variables set either by an exchange or a regulatory body, we may
investigate the impact of leverage and short-selling limits on the arbitrage activity. From
the characterization in Proposition 4.1, we see that increasing these limits ðg;bÞ shrinks the
mispricing region, and hence makes mispricing less likely to occur. Furthermore,
increasing these limits increases the interest rate, moving it closer to the perfect risk-
sharing case, and reduces the magnitude of the mispricing and thus the arbitrageur’s
profits: Overall, there is less arbitrage activity. This is because increasing these limitsalleviates the risky investment restrictions on the investors, and hence the two investors are
better able to share risk (with or without the arbitrageur). This in turn provides for a
smaller role for the arbitrageur, reducing the mispricing and the arbitrageur’s profits.
4.3. The arbitrageur and risk-sharing
To clarify the role of the arbitrageur, we quantify the transfers of risk that take place in
region ðc; eÞ, as well as in our benchmark economies I and II. We will additionally consider
the following benchmark case:Economy III: no securities (the only investment opportunity is the stock).
In Economy III, investors do not trade and they invest all of their wealth in the stock
ðyi S ¼ W i Þ.
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We introduce Fi , investor i ’s measure of risk exposure, defined as the proportion of his
wealth held in risky investments,
Fi ðtÞ yi
S ðtÞ þ yi P ðtÞ
W i ðtÞ. (38)
In the absence of trade (Economy III), all of agent i ’s wealth is invested in the stock, and so
Fi III ¼ 1 for both investors. When trade is possible, heterogeneity in beliefs leads the more
optimistic investor to optimally take on a higher risk exposure than in the no-trade case,
and the more pessimistic agent to take on a lower risk exposure. Corollary 4.1 reports the
investors’ equilibrium risk exposures both in the absence of binding constraints
(benchmark economy I and region ða; aÞ of our economy) and in the equilibrium case
ðc; eÞ of our economy.
Corollary 4.1. In the absence of binding constraints, equilibrium risk exposures are
F1I ðtÞ ¼ 1 þ ð1 À lðtÞÞ
¯ mðtÞ
sand F2
I ðtÞ ¼ 1 À lðtÞ¯ mðtÞ
s. (39)
In the ðc; eÞ equilibrium,
F1ðtÞ ¼ b þ g1 À lðtÞ
lðtÞþ
M
lðtÞW ðtÞand
F2ðtÞ ¼ 1 À g À ðb À 1ÞlðtÞ
1 À lðtÞÀ
M
ð1 À lðtÞÞW ðtÞ. ð40Þ
In the presence of constraints, once the disagreement ð ¯ mÞ is sufficiently large for theconstraints to bind, the amount of risk that is traded becomes ‘‘stuck’’: Portfolio holdings
do not depend on ¯ m any more. It is straightforward to verify that, assuming the conditions
for ðc; eÞ to occur in our economy are fulfilled,
F1IIIðtÞoF1
IIðtÞoF1ðtÞoF1I ðtÞ; and F2
IIIðtÞ4F2IIðtÞ4F2ðtÞ4F2
I ðtÞ. (41)
Thus, the presence of the arbitrageur makes the allocation of risk closer to what it would
be without constraints. A natural measure of the amount of risk that is shared is the
difference F1 À F2, which is equal to zero in the no-trade case (benchmark Economy III)
and, in the absence of binding constraints (benchmark economy I and region ða; aÞ in oureconomy), grows linearly with heterogeneity (it is then equal to ¯ m=s). In our equilibrium
ðc; eÞ,
F1ðtÞ À F2ðtÞ ¼b À 1
1 À lðtÞþ
g
lðtÞþ
M
W ðtÞlðtÞð1 À lðtÞÞ, (42)
revealing how the amount of risk that is shared between the investors grows with the size of
the arbitrageur’s position. The mechanism through which the arbitrageur facilitates risk-
sharing between the investors is the one already described in Section 2: The arbitrageur
effectively acts as a financial intermediary who acts as an additional counterparty to the
investors. By converting stock positions into derivative positions, the arbitrageur allowsthe investors to trade more with each other. The analysis in Section 2 can be repeated with
only insubstantial changes, provided one replaces investors’ state-o cashflows (Ai s) in
Section 2 with their risk exposures (Fi s). In the absence of the arbitrageur (benchmark
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Economy II), investors’ risk exposures are
F1IIðtÞ ¼ b þ g
1 À lðtÞ
lðtÞand F2
IIðtÞ ¼ 1 À g À ðb À 1ÞlðtÞ
1 À lðtÞ. (43)
Even though the optimistic investor 1 would like to increase his risk exposure (to F1I ) by
increasing his holding of the derivative (since he is already leverage-constrained on the
stock), this is impossible because then investor 2 would not be able to act as a counterparty
due to his lower constraint on P . By shorting the derivative, the arbitrageur provides an
extra counterparty, which allows for an increase in F1 (by M =lW ) that is proportional to
the arbitrageur’s position limit M . Similarly, it is impossible for the pessimistic investor 2
to invest as little in the stock as he optimally would, because then not all of the stock would
be held and so markets would not clear. By investing in the stock, the arbitrageur makes it
possible for the pessimistic investor to hold less of it and thereby reduce his risk exposure
(by M =½ð1 À lÞW ). The introduction of the arbitrageur in the economy is equivalent to
that of an extra, zero-net-supply security (e.g., a total return swap on the stock), in which
both investors face a position limit proportional to M . This security is purchased from the
pessimistic investor by the innovator, the arbitrageur, and resold to the optimistic investor
at a higher price (or a higher notional riskless rate, in the case of the swap). The difference,
the mispricing, makes up the arbitrageur’s profit.
Finally, we note that much of the above analysis is not dependent on the logarithmic
utility assumption. The investors’ portfolio holdings in region ðc; eÞ follow directly from the
binding constraints and the market clearing conditions. Thus, whenever mispricing occurs,
these holdings obtain for arbitrary investor preferences or beliefs, and our analysis of the
arbitrageur’s contribution to improved risk-sharing is equally valid. Only the occurrence of mispricing is needed for this to follow. Basak and Croitoru (2000, Proposition 5) show how
and when mispricing may arise in equilibrium under very general assumptions,
demonstrating the robustness of our analysis.
4.4. Comparison with an economy with three investors and no arbitrageur
It is well known that, ceteris paribus, increasing the number of agents typically improves
risk-sharing in an economy.2 Thus, naturally the question arises as to whether adding a
third (risk-taking) investor to our economy would improve risk-sharing in the same way
that the arbitrageur does. Proposition 4.2 presents an example of equilibrium that can arisein this case. We index the third investor (logarithmic expected utility maximizer) by i ¼ 3Ã
and, accordingly, denote his wealth by W 3Ã and his estimate of mS by m3ÃS . As before,
W W 1 þ W 2 and l W 1=W .
Proposition 4.2. In the presence of a third optimizing agent ð3ÃÞ, but no arbitrageur, if
¯ mðtÞ4sðb À 1Þ1 þ
W 3ÃðtÞW ðtÞ
1 À lðtÞþ gs
1 þW 3ÃðtÞ
W ðtÞ
lðtÞ þW 3ÃðtÞ
W ðtÞ
þm1
S ðtÞ À m3ÃS ðtÞ
s
W 3ÃðtÞ
W ðtÞ, (44)
ARTICLE IN PRESS
2For example, in a standard economy the risk aversion of the representative agent A satisfies 1=A ¼ Si ð1=Ai Þ
and so decreases when the number of agents in the economy increases, ceteris paribus. Hence, the market price of
risk, Asd, where sd is the volatility of aggregate consumption, and the individual consumption volatilities, Asd=Ai ,
also decrease.
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Àgs 1 þW ðtÞ
W 3ÃðtÞ
pm1
S ðtÞ À m3ÃS ðtÞ
spgs
1 þ W 3ÃðtÞW ðtÞ
lðtÞ, (45)
and
1 À lðtÞ
lðtÞ þ W 3ÃðtÞW ðtÞ
Xb À 1, (46)
then in equilibrium investors are in optimization cases ðc; e; cÞ and the risk exposures of
investors 1 and 2 are
F1ÃðtÞ ¼ bs þ gs1 À lðtÞ
lðtÞ þ W 3ðtÞW ðtÞ
þm1
S ðtÞ À m3ÃS ðtÞ
s
W 3ÃðtÞ
lðtÞW ðtÞ þ W 3ÃðtÞ, (47)
and
F2ÃðtÞ ¼ sð1 À gÞ À ðb À 1ÞslðtÞ þ W 3ÃðtÞ
W ðtÞ
1 À lðtÞ. (48)
Investors’ equilibrium risk exposures reveal that a third risk-taking investor affects risk-
sharing differently than the arbitrageur does. While F2IoF
2ÃoF2
II, implying that the
presence of the extra investor 3Ã makes investor 2’s risk exposure closer to its value in an
unconstrained economy ðF2I Þ, the effect on investor 1 is ambiguous. Depending on investor
3Ã’s beliefs, F1Ã can be either higher or lower than if the third investor were not present.
Accordingly, the effect of investor 3Ã on the amount of risk shared between investors 1 and
2 is ambiguous. Our measure of risk-sharing is now equal to
F1ÃðtÞ À F2ÃðtÞ ¼ ðb À 1Þs1 þ
W 3ÃðtÞW ðtÞ
1 À lðtÞþ gs
1 þW 3ÃðtÞ
W ðtÞ
lðtÞ þ W 3ÃðtÞW ðtÞ
þm1
S ðtÞ À m3ÃS ðtÞ
s
1
1 þ lðtÞ W ðtÞ
W 3ÃðtÞ
. ð49Þ
Compared to the value of this measure in the absence of binding constraints (which
remains equal to ¯ m, whether or not a third investor is present), if investor 3 Ã is sufficiently
optimistic (while still allowing for the conditions in the proposition to hold), risk-sharingcan be degraded by his presence. An example of plausible parameters for which this
happens is as follows: g ¼ 0:5, b ¼ 1:5, sS ¼ sP ¼ 0:1, lðtÞ ¼ 0:5, W 3ÃðtÞ=W ðtÞ ¼ 0:4,
m1S ðtÞ ¼ 0:1, m2
S ðtÞ ¼ 0:05, and m3ÃS ðtÞ ¼ 0:105. Then, the equilibrium is as described in
Proposition 4.2, and the amount of risk shared between investors 1 and 2 is smaller than if
the third investor were not present: F1 À F2 is equal to 0.1761 versus 0.2 (and 0.5 in an
unconstrained economy, whether or not a third investor is present).
Thus, our arbitrageur, whose presence always improves risk-sharing across investors (as
demonstrated by (42)), performs a specific economic function.
5. Equilibrium with a noncompetitive arbitrageur
In this section, we consider another imperfection for the financial markets, in that the
arbitrageur has market power therein. This could naturally arise in markets with limited
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liquidity or in specialized markets with few participants. There is considerable evidence
(see, e.g., Attari and Mello, 2002) suggesting that in such markets, large traders ‘‘move’’
prices.
When the arbitrageur has market power in the securities markets, he takes account of
the price impact of his trades on the level of mispricing across the risky assets. Thisconsideration is shown to induce much richer arbitrageur trades than those in the
competitive case, in which under mispricing, the arbitrageur simply takes on the largest
position allowed by his investment restriction. The importance of this noncompetitive case
is further underscored by the fact that trading restrictions on the arbitrageur are no longer
needed to bound arbitrage, and hence attain equilibrium with an active arbitrageur, as
demonstrated in this section.
5.1. The noncompetitive arbitrageur
For tractability, we consider a myopic arbitrageur in the sense that he maximizes his
current profits (as opposed to lifetime profits). This shortsightedness may be interpreted as
capturing in reduced form arbitrageurs that are either short-lived or have short-termism, as
often observed in practice. This may, for example, be due to the separation of ‘‘brains and
resources’’ that prevails in real-life professional arbitrage (Shleifer, 2000, Chapter 4): If the
arbitrageur reports to shareholders who are less sophisticated than himself, he may be
enticed to choose short-term profits over a long-term strategy that may return less in the
short run.
An arbitrageur with market power faces a problem that is different—and more
complex—than the competitive arbitrageur of Section 4. The noncompetitive arbitrageurobserves the investors’ demands as functions of prices (as determined in Section 3), and
then chooses the size of his own trades so as to maximize his concurrent profits by taking
the exact counterposition of the investors’ trades so that markets clear. Of course, he will
only be active when there is mispricing and hence positive profits to be made; otherwise his
profits are zero. Accordingly, given the investors’ optimal investments (Proposition 3.1),
clearing in the security markets implies the following when mispricing (case ðc; eÞ) occurs:
DS ;P ðtÞ ¼ ¯ mðtÞs Àðb À 1Þs2
1 À lðtÞÀgs2
lðtÞÀ
y3S ðtÞs2
lðtÞð1 À lðtÞÞW ðtÞ. (50)
Hence, the arbitrageur’s influence on security prices manifests itself, via (50), as the size of his stock position ðy3
S Þ that affects mispricing. The larger his position, the lower the
mispricing—in a sense, the prices move against him: The value of his marginal profit (the
mispricing) is reduced by his trades.
The noncompetitive arbitrageur solves the following optimization problem at time t:
maxy3
S ðtÞ;DS ;P ðtÞ
C3ðtÞ ¼ y3S ðtÞDS ;P ðtÞ (51)
s.t. y3S ðtÞ þ y3
P ðtÞ þ y3B ðtÞ ¼ 0; ðy3
S ðtÞ þ y3P ðtÞÞ s ¼ 0; (21)2(22)
and DS ;P ðtÞ satisfies (50).The arbitrageur solves simultaneously for the optimal size of his arbitrage position and the
equilibrium mispricing, given his understanding of how the equilibrium mispricing
responds to his trades (50). As for the competitive arbitrageur, the costless, riskless
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arbitrage conditions, (21)–(22), uniquely determine the noncompetitive arbitrageur’s
holdings in financial securities in terms of his stock investment. Finally, we note that the
maximization problem (51) is a simple, concave quadratic problem.
5.2. Analysis of equilibrium
Proposition 5.1 reports the possible equilibrium cases and the characterization for each
case.
Proposition 5.1. The equilibrium with a noncompetitive arbitrageur is as follows.
When ¯ mðtÞpgs=lðtÞ þ min s=lðtÞ; ðb À 1Þs=1 À lðtÞÈ É
; investors are in ða; aÞ, and when
¯ mðtÞ4ðg þ 1Þs=lðtÞ and lðtÞX1=b, investors are in ða; bÞ. In both cases, DS ;P ðtÞ, rðtÞ, and lðtÞ
are as in the competitive arbitrageur equilibrium described in Proposition 4.1.
When ¯ mðtÞ4gs=lðtÞ þ sðb À 1Þ=1 À lðtÞ and lðtÞo1=b; investors are in ðc; eÞ and the
arbitrageur’s equilibrium stock investment and profit are given by
y3S ðtÞ ¼
lðtÞð1 À lðtÞÞW ðtÞ
2s¯ mðtÞ À
sðb À 1Þ
1 À lðtÞÀ
gs
lðtÞ
!(52)
and
C3ðtÞ ¼lðtÞð1 À lðtÞÞW ðtÞ
4¯ mðtÞ À
sðb À 1Þ
1 À lðtÞÀ
gs
lðtÞ
!2
, (53)
respectively, and the equilibrium mispricing, interest rate, and distribution of wealth dynamics
are given by
DS ;P ðtÞ ¼1
2¯ mðtÞs À
ðb À 1Þs2
1 À lðtÞÀgs2
lðtÞ
!, (54)
rðtÞ ¼ m2S ðtÞ À s2 þ s ¯ mðtÞ
lðtÞ
2þ
1
2gs2 þ
1
2
lðtÞ
1 À lðtÞs2ðb À 1Þ, (55)
and
dlðtÞ ¼ lðtÞ DS ;P ðtÞ ðb À 1Þ þy3
S ðtÞ
W ðtÞ
!þ ½ð1 À lðtÞÞð ¯ mðtÞ À DS ;P ðtÞ=sÞ2
& 'dt
þ lðtÞð1 À lðtÞÞð ¯ mðtÞ À DS ;P ðtÞ=sÞ dw1ðtÞ, ð56Þ
respectively. In all cases, the aggregate wealth dynamics follow:
dW ðtÞ ¼ W ðtÞ m1S ðtÞ À
1
T À t
À y3
S ðtÞDS ;P ðtÞ
!dt þ W ðtÞsdw1ðtÞ. (57)
As in the competitive case, three cases are possible in equilibrium, with case ðc; eÞ being
the mispricing case. Comparison with the competitive equilibrium of Proposition 4.1
reveals that the region of mispricing is larger in the noncompetitive equilibrium. Hence, the
presence of the noncompetitive arbitrageur makes mispricing more likely to occur. This is
intuitive: in the presence of mispricing, the competitive arbitrageur always trades to the fullextent allowed by the position limit, and thus is more likely to fully ‘‘arbitrage away’’ the
mispricing, making it disappear. In the noncompetitive case, in contrast, he limits his
trades so as to make the mispricing (and positive arbitrage profits) subsist under a broader
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range of conditions. It is impossible for the arbitrageur to make the mispricing case more
likely because, whenever he trades, he improves risk-sharing across investors and reduces
the mispricing. As a result, unlike in the competitive case, the conditions for regions ðc; eÞ
are as if the arbitrageur were not present.
As compared with the no-arbitrageur economy (Economy II, Proposition 4.1 withM ¼ 0), we see that the equilibrium mispricing is reduced by a half, DS ;P ðtÞ ¼ ð1=2ÞDII
S ;P ðtÞ.
That is, instead of providing a perfect counterparty to the two investors (which would
eliminate the mispricing), when the arbitrageur is a non-price-taker, he provides only one-
half of a perfect counterparty. This is also easy to understand: The arbitrageur’s profit is
proportional to the product of the mispricing and the size of his position. The mispricing is
affine in y3S , so the profit is a quadratic function of the arbitrageur’s position, and it is
therefore maximized at a position halfway between zero (which would lead to zero profit)
and a perfect counterparty to the investors (which would lead to zero mispricing, and
hence zero profit).
In contrast to the constant competitive case, the noncompetitive arbitrageur’s stock
position is stochastic, driven by the investors’ difference of opinion ¯ m, the investors’
aggregate wealth W , and the distribution of wealth l. For sufficiently low investor
disagreement, the noncompetitive arbitrageur’s stock position is lower than in the
competitive case ðy3S ðtÞoM Þ, and his position is higher for high investor disagreement.
Hence, for low investor disagreement, the noncompetitive equilibrium mispricing is higher
than the competitive one. Consequently, the economic role of the arbitrageur (as discussed
in Section 4), in terms of alleviating the effect of portfolio constraints and facilitating trade
amongst investors, is reduced. The opposite holds when there is high disagreement
amongst investors: The economic role of the noncompetitive arbitrageur is morepronounced. Nonetheless, our discussion of the economic role of the arbitrageur in
Section 4 remains valid when the exogenous bound on the arbitrageur’s position M is
replaced with the endogenous y3S . Finally, in contrast to the linear competitive case, the
noncompetitive arbitrageur’s profits are convex in the amount of disagreement amongst
investors.
6. Equilibrium with an arbitrageur subject to margin requirements
We now return to a competitive market and assume that the arbitrageur is subject to
margin requirements. The arbitrageur then needs to be endowed with some capital, whichwe assume is held as an investment by the investors. In addition to being more realistic, this
modification of our model allows us to endogenize the amount of capital allocated to
arbitrage activity and the size of the arbitrage positions.
6.1. The arbitrageur under margin requirements
The economic setup introduced in Sections 3 and 4 is modified as follows. Riskless,
costless storage, henceforth referred to as cash, is available to the agents in addition to the
stock, derivative, and bond. Assuming rðtÞ40, the investors would never find it optimal to
invest in cash. This assumption is verified to hold in equilibrium for an appropriate choiceof the exogenous parameters, including the pessimistic investor’s prior belief distribution
on mS . However, the arbitrageur may be forced to hold cash due to his margin
requirements, that are as follows: letting y3C and W 3 denote the arbitrageur’s cash holding
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and capital, respectively, his holdings must obey
W 3ðtÞXZðyS ðtÞ þ jy3P ðtÞjÞ (58)
and
y3C ðtÞXð1 þ ZÞ maxfÀy3
P ðtÞ; 0g, (59)
where Z; 2 ½0; 1 and y3S ðtÞX0 (hence, there is no need to account for short positions in S
in (58)–(59)). While Eq. (58) limits the size of the arbitrageur’s position in proportion to his
capital, Eq. (59) states that the arbitrageur does not enjoy the use of proceeds from his
short position in the derivative; rather, these must be kept in cash as a deposit. In addition,
a fraction of the margin on the short sale must also be met with cash. For clarity, we refer
below to (58) as the ‘‘margin constraint’’ and to (59) as the ‘‘cash constraint.’’ Our
modeling of margin requirements is standard (see, e.g., Cuoco and Liu, 2000). This
modeling of margin requirements could easily be amended without affecting our main
intuition. All that is really needed is the presence of a constraint that limits the size of the
arbitrageur’s position in proportion to his capital, and that short sales be costly. For
simplicity, we assume that there is no difference between initial and maintenance margins.
Denoting y3 y3B ; y
3S ; y
3P ; y
3C
À Á, the arbitrageur’s problem is
maxy3
E 3Z T
0
C3ðtÞ dt
!(60)
s.t. C3ðtÞ dt ¼ fy3S ðtÞm3
S ðtÞ þ y3P ðtÞm3
P ðtÞ þ y3B ðtÞrðtÞg dt
þ ½y
3
S ðtÞ þ y
3
P ðtÞsdw
3
ðtÞ, ð61Þ
y3S ðtÞ þ y3
P ðtÞ þ y3B ðtÞ þ y3
C ðtÞ ¼ W 3ðtÞ, (62)
ðy3S ðtÞ þ y3
P ðtÞÞ s ¼ 0, (63)
and ð58Þ À ð59Þ hold.
Both the amount of arbitrage capital ðW 3Þ and its return ðC3ðtÞ=W 3ðtÞÞ are determined
endogenously in equilibrium. Restrictions (62) and (63) and the equality in (59) (which
always holds because cash is dominated by the bond) yield all holdings as a function of
y3P ðtÞ:
y3S ðtÞ ¼ Ày3
P ðtÞ; y3B ðtÞ ¼ Àð1 þ ZÞ maxfÀy3
P ðtÞ; 0g, (64)
and
y3C ðtÞ ¼ ð1 þ ZÞ maxfÀy3
P ðtÞ; 0g. (65)
When there is no mispricing ðDS ;P ðtÞ ¼ 0Þ, the arbitrageur is indifferent between all portfolios
satisfying (58), (64)–(65), and y3P ðtÞX0; the last inequality holds because (59) penalizes short
sales (the arbitrageur loses the interest on his cash deposit). As before, all feasible portfolio
holdings yield zero profit. In the presence of mispricing ðDS ;P ðtÞ40Þ, it is optimal for the
arbitrageur to take on as large a profitable arbitrage position (long in the cheap stock, and
short in the more expensive derivative) as feasible; hence, (58) holds with equality:
Zðy3S ðtÞ À y3
P ðtÞÞ ¼ W 3ðtÞ, (66)
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distribution of wealth dynamics, and aggregate wealth dynamics are
DS ;P ðtÞ ¼ ¯ mðtÞs Àðb À 1Þs2
1 À lðtÞÀgs2
lðtÞÀ
W 3ðtÞs2
lðtÞð1 À lðtÞÞW ðtÞ
lðtÞð1 þ ZÞ þ 1
2Z
40, (74)
rðtÞ ¼ m2S ðtÞ À ð1 À gÞs2 þ
ðb À 1ÞlðtÞ
1 À lðtÞþ
W 3ðtÞ
ð1 À lðtÞÞW ðtÞ
2 þ Z
2Z
!s2, (75)
dlðtÞ ¼ fDS ;P ðtÞðb À lðtÞÞ þ ðs þ slðtÞÞ½sð1 À 2lðtÞÞ þ slðtÞ þ lðtÞs2g dt
þ lðtÞslðtÞ dw1ðtÞ, ð76Þ
where
slðtÞ ¼ ð1 À lðtÞÞð ¯ mðtÞ À DS ;P ðtÞ=sÞ ÀW 3ðtÞð1 þ ZÞs
2ZW ðtÞ
, (77)
and
dW ðtÞ ¼ W ðtÞ À ð1 þ ZÞ gð1 À lðtÞÞW ðtÞ þW 3ðtÞ
2Z
!& 'ðm1
S ðtÞ dt þ sdw1ðtÞÞ
ÀW ðtÞ
T À tdt. ð78Þ
Because arbitrage is riskless, equilibrium is only possible if the rate of return on
arbitrage capital is equal to the riskless interest rate; otherwise, investors would face an
unbounded arbitrage opportunity and equilibrium would not obtain. From (69), thisimplies that DS ;P ðtÞ ¼ rðtÞð1 þ ZÞ, which leads (using (74) and (75)) to an equation affine in
W 3ðtÞ whose solution is given by (73). The endogenous amount of arbitrage capital W 3 is
provided by Eq. (73). Interestingly, the main factor that drives the equilibrium amount of
arbitrage activity is similar to the noncompetitive case of Section 5: The amount of
arbitrage is driven by the mispricing that would prevail without an arbitrageur,
DIIS ;P ¼ ¯ ms À ðb À 1Þs2=ð1 À lÞ À gs2=l, i.e., the amount of heterogeneity that cannot be
traded by the investors due to their constraints. This is intuitive: the higher this untraded
heterogeneity, the higher the profit opportunities for the arbitrageur/financial inter-
mediary. In addition, the amount of arbitrage capital is increasing in the severity of the
margin constraint (58) (as measured by Z): the more stringent the constraint, the higherthe amount of capital needed to achieve a similar result. In contrast, W 3 is decreasing in
the severity of the cash constraint (59) (i.e., in ð1 þ ZÞ). The rationale for this is clarified by
our discussion on the respective impact of these two constraints on risk-sharing.
The structure of equilibrium closely resembles that of Section 4, with the exogenous
position size M replaced by the endogenous arbitrageur’s stock investment y3S ¼ W 3=2Z.
Nonetheless, risk-sharing between investors is further impacted by the cash constraint.
This is evidenced by the diffusion of the wealth distribution dynamics, sl, which exhibits
an additional term proportional to ð1 þ ZÞ. The expressions for the interest rate and the
mispricing include a similar term, suggesting that the more severe the cash constraint, the
better risk-sharing between investors.To investigate risk-sharing between investors, we examine the investors’ risk exposures,
as provided in Proposition 6.2. We note that the expressions in the proposition are not
dependent of our assumption of logarithmic utility for the investors, as the expressions
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follow from the investors’ binding constraints, the arbitrageur’s policy (67)–(68), and
market clearing. (They would also hold in a pure exchange economy.)
Proposition 6.2. In the mispriced equilibrium with an arbitrageur subject to the margin
requirements (58)–(59), the investors’ risk exposures are
F1ðtÞ ¼ b þ g1 À lðtÞ
lðtÞþ
W 3ðtÞ
2ZlðtÞW ðtÞ, (79)
and
F2ðtÞ ¼ ð1 À gÞ À ðb À 1ÞlðtÞ
1 À lðtÞÀ
ð2 þ ZÞW 3ðtÞ
2Zð1 À lðtÞÞW ðtÞ. (80)
Eq. (80) reveals that the pessimistic investor’s equilibrium risk exposure is decreased by
the severity of the cash constraint (measured by ), which leads to an improvement in risk-
sharing due to the cash constraint. This is because the cash deposit is taken out of theaggregate amount of the good to be invested in production. Hence, market clearing
requires y1S þ y2
S þ y3S ¼ W 3 À y3
C . Thus, the higher the arbitrageur’s cash holding, the less
the pessimistic investor has to invest in the stock for markets to clear. (Investor 2’s
investment has to adjust to clear the markets because investor 1’s leverage constraint on
the stock is binding.) This makes it possible for the pessimistic investor to have a lower risk
exposure, closer to the value it would have in an unconstrained equilibrium. Investor 1’s
risk exposure is unaffected by the arbitrageur’s cash constraint, thereby leading to a higher
value for our measure of risk-sharing ðF1 À F2Þ. Thus, the cash constraint mitigates the
effect of the margin constraint, albeit in an indirect way. From (80), the effect of the cash
constraint on investor 2’s risk exposure is equivalent to an increase in the amount of
arbitrage activity. The effect of the margin constraint (58), on the other hand, is as would
be expected: It is similar to that of the arbitrageur’s investment restriction in Section 4, and
the intuition therein follows. In fact, in the absence of any cash requirement ð1 þ Z ¼ 0Þ,
the expressions from the mispriced equilibrium in Section 4 apply once M is replaced with
the endogenous y3S .
The model of equilibrium arbitrage activity in this section is admittedly simplistic in
many respects. In particular, our implication that the return on arbitrage capital equals the
riskless interest rate is undoubtedly unrealistic. Nonetheless, we believe that the general
approach of this section is consistent with the nature of arbitrage in contemporaryfinancial markets, with specialized agents engaging in arbitrage, such as hedge funds,
whose capital is held by outside investors, and could be extended to a more realistic setup.
Finally, we demonstrate the different ways in which the constraints imposed on the
arbitrageur can affect risk-sharing.
7. Conclusion
In this article, we attempt to shed some light on the potential role of arbitrageurs in
rational financial markets. Towards that end, we develop an economic setting with
heterogeneous risk-averse investors subject to leverage and short-sales restrictions, and anarbitrageur engaging in costless, riskless arbitrage trades. In the presence of arbitrage
opportunities, the arbitrageur is shown to improve risk-sharing amongst investors. The
arbitrageur effectively plays the role of a financial intermediary, or an innovator
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synthesizing a derivative security specifically designed to relieve investors’ restrictions. The
improved transfer of risk due to the arbitrageur is shown to be equally valid when the
arbitrageur behaves noncompetitively or is subject to margin requirements and needs
capital to implement his arbitrage trades. Although we make simplifying assumptions on
the primitives of the economy (preferences, investment opportunities, and investmentrestrictions) for tractability, our insights can be readily extended to more general primitives
and to a pure exchange economy.
Our framework is amenable to the study of further related issues. For example, the
analysis of the arbitrageur under margin requirements could be extended to make arbitrage
activity risky by allowing for the possibility of volatility jumps in a discretized version of
the model. Then, the risk-return tradeoff of arbitrage capital being consistent with asset
prices would pin down the endogenous amount of arbitrage capital. Our model also sheds
new light on the issue of who the users of derivatives markets such as futures are, what the
motivations of the different categories of market participants are, and what returns each
participant type obtains. Typically, the empirical literature on futures markets (e.g.,
Ederington and Lee, 2002; Piazzesi and Swanson, 2004), distinguishes between hedgers and
speculators. Our model underscores the need to additionally account for the activity of
arbitrageurs, and provides theoretical tools to understand how their positions and their
returns are related to underlying economic variables.
Appendix A. Proofs
Proof of Proposition 3.1. The investors’ optimization problem is nonstandard in several
respects: the portfolio constraints, the redundancy in the risky investment opportunities(technology and stock), and the mispricing. The solution technique, developed by Basak
and Croitoru (2000), involves using investors’ nonsatiation (implying that, in the presence
of mispricing, the investment restrictions always bind) to convert the original problem with
two risky investment opportunities into one with a single, fictitious risky asset with
nonlinear drift. Techniques of optimization in nonlinear markets (Cvitanic and Karatzas,
1992) can then be applied. &
Proof of Proposition 4.1. Combining the agents’ optimal policies (Proposition 3.1 and
Section 4.1) and the conditions for market clearing, and noting that both investors face the
same price process for the derivative P , implies
m1P ðtÞ À m2
P ðtÞ
sdt ¼ dw2ðtÞ À dw1ðtÞ ¼
m1S ðtÞ À m2
S ðtÞ
sdt; hence
m1P ðtÞ À m2
P ðtÞ
s¼ ¯ mðtÞ, ðA:1Þ
and leads to the equilibrium expressions for r and DS ;P in each case. Substituting these
equilibrium prices in the conditions for the investors’ optimization cases leads to the
conditions for the equilibrium case ðc; eÞ and the first condition for ða; bÞ. The conditions
for ða; aÞ and the second condition for ða; bÞ are those under which it is possible to find
portfolio holdings that, among all those between which agents are indifferent, clearmarkets.
To verify the conditions for the ða; aÞ case, observe that if the economy is in
ða; aÞ, each of the three agents is indifferent among all the portfolio holdings that
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satisfy, respectively,
y1S ðtÞ
W 1ðtÞþ
y1P ðtÞ
W 1ðtÞ¼m1
S ðtÞ À rðtÞ
s2¼ 1 þ
¯ mðtÞð1 À lðtÞÞ
s, (A.2)
y2S ðtÞ
W 2ðtÞþ
y2P ðtÞ
W 2ðtÞ¼m2
S ðtÞ À rðtÞ
s2¼ 1 À
¯ mðtÞlðtÞ
s, (A.3)
y3P ðtÞ ¼ Ày3
S ðtÞ and y3B ðtÞ ¼ 0, (A.4)
where the equilibrium value for the interest rate has been substituted into the investors’
portfolio demands. In addition, agents’ portfolio holdings must satisfy the clearing
conditions that are equivalent to
y1
S ðtÞ þ y2
S ðtÞ þ y3
S ðtÞ ¼ W 1
ðtÞ þ W 2
ðtÞ; and y1
P ðtÞ þ y2
P ðtÞ þ y3
P ðtÞ ¼ 0. (A.5)
There exist an infinity of solutions to the system formed by Eqs. (A.2)–(A.5). The
conditions for case ða; aÞ provided in the proposition ensure that there exists a solution that
satisfies the agents’ position limits (11) and (23). To verify this, express the solutions of the
system as a function of two free parameters (say, y1P ðtÞ and y3
S ðtÞ), since there are two
degrees of freedom in the system, and substitute these expressions into the position limits.
This results in eight inequalities that y1P ðtÞ and y3
S ðtÞ must obey. The conditions for case
ða; aÞ ensure that there is no contradiction among these, so that there exist portfolio
holdings that satisfy all the requirements for the unconstrained equilibrium ða; aÞ. The
second condition for case ða;
bÞ can be obtained in a similar fashion.
It is easy to verify that no other cases are possible in equilibrium (the only other cases
compatible with market clearing, (b, a) and (e, c), are impossible given that investor 1 is
more optimistic). Substituting the prices and optimal policies into the investors’ dynamic
budget constraints and using Ito’s Lemma leads to the dynamics of l and W , while the
expressions for the investors’ consumptions follow from Proposition 3.1. &
Proof of Corollary 4.1. The expressions follow from the definition of Fi , the optimal
policies in Proposition 3.1, and the equilibrium prices of Proposition 4.1. &
Proof of Proposition 4.2. The expressions follow from the optimal policies in Proposition
3.1 and market clearing. Substituting the equilibrium prices into the conditions for theoptimality cases leads to the conditions for the ðc; e; cÞ equilibrium. Details are omitted for
brevity. &
Proof of Proposition 5.1. In the presence of an active arbitrageur, market clearing and
agents’ optimal policies lead to the expressions for the equilibrium mispricing as a function
of the arbitrageur’s position (50). Substitution into the arbitrageur’s optimization problem
(51) leads to a quadratic problem whose solution is provided by (52). Substitution into the
problem’s objective function leads to the optimal value of C3ðtÞ. Given the arbitrageur’s
position, the equilibrium values of the mispricing, interest rate, distribution of wealth
dynamics, and aggregate wealth dynamics are deduced in a fashion similar to thecompetitive case (Proposition 4.1). The conditions for the equilibrium are those under
which there exists a solution to the arbitrageur’s optimization such that he realizes a
positive profit, and the investors are optimally in case (c, e). &
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Proof of Proposition 6.1. The interest rate and mispricing, as a function of W 3, follow
from the arbitrageur’s holdings in (67)–(68), the investors’ demands (Proposition 3.1), and
market clearing in a fashion similar to Proposition 4.1. To pin down the amount of
arbitrage capital W 3, observe that there only exists a solution to the investors’
optimization if the (riskless) return on arbitrage (69) equals the riskless rate r; otherwise,the investors would face an unbounded arbitrage opportunity. By substituting the interest
rate and mispricing into the expression for the return on arbitrage capital (69), we obtain
(73), the only amount of arbitrage capital such that this restriction holds. The conditions in
the proposition ensure that the aggregate amount invested in arbitrage activity is
nonnegative, and that the investors, given the trades of the arbitrageur, are in cases (c)
and (e). &
Proof of Proposition 6.2. The investors’ portfolio holdings can be deduced from their
binding constraints, the arbitrageur’s holdings (67)–(68), and market clearing. Applying
the definition of Fi then yields the expressions in the proposition. &
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